A multifractal boundary spectrum for $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$


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Abstract

jats:titleAbstract</jats:title>jats:pWe study jats:inline-formulajats:alternativesjats:tex-math$${{,\mathrm{SLE},}}_\kappa (\rho )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:msub mml:mrow <mml:mspace /> mml:miSLE</mml:mi> <mml:mspace /> </mml:mrow> mml:miκ</mml:mi> </mml:msub> mml:mrow mml:mo(</mml:mo> mml:miρ</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> curves, with jats:inline-formulajats:alternativesjats:tex-math$$\kappa $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miκ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternativesjats:tex-math$$\rho $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miρ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed “angle” and determine the almost sure Hausdorff dimensions of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps jats:inline-formulajats:alternativesjats:tex-math$$g_t$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mig</mml:mi> mml:mit</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate. </jats:p>

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Funder: University of Cambridge

Keywords
4902 Mathematical Physics, 49 Mathematical Sciences
Journal Title
Probability Theory and Related Fields
Conference Name
Journal ISSN
0178-8051
1432-2064
Volume Title
178
Publisher
Springer Science and Business Media LLC